How do you change a mixed number to an improper fraction?
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A thumb-wrestling match requires two thumbs, so we can either suppose that we need at least two people 
, or allow one to thumb-wrestle one's self. In either case, we'd have 
, so let's just say we need a minimum of two players.
If we add one more person to the set of players, then the first two people would need to play 1 additional match each. So
.
If we add one more person, then the first three people would again each have to play 1 more match with the new person. So
.
And so on, so that in general, the number of games needed for everyone to play exactly one match with everyone else is given recursively by
If we add one more person to the set of players, then the first two people would need to play 1 additional match each. So
If we add one more person, then the first three people would again each have to play 1 more match with the new person. So
And so on, so that in general, the number of games needed for everyone to play exactly one match with everyone else is given recursively by